Some Misconceptions In A PBS Einstein Biography
- From: markwh04@xxxxxxxxx
- Date: Fri, 16 Dec 2005 07:39:56 +0000 (UTC)
A while ago, a biographical series on Einstein was presented on PBS
which reiterated a widely-held Physics folk-myth regarding the relation
of Einstein and Maxwell's equations.
Whatever route of inquiry Einstein took to get to his ideas, the one
thing that is clear is that it did not come about by reading the
"invariance of light speed" notion into Maxwell's treatise or the
equations that Maxwell actually wrote down.
Though it is true that the equations that are called "Maxwell's
equations" are Poincare' invariant and lead directly to the precepts of
Relativity, in fact Maxwell's equations (the ones present in Maxwell's
treatise) are NOT Poincare' invariant, but Galilean invariant. This is,
in fact, part of a larger story of what happened to the "C" and "G" in
the litanny of alphabet soup that only comes down to the present as A,
B, D, E, F (force density), H, I (magnetization) and J.
The actual equations include the ones known today as the macroscopic
equations:
div D = rho; curl H = C = J + dD/dt; div B = 0; curl
E - dB/dt = 0
and the constitutive relations, which in modern notation, read:
D = epsilon E; B = mu H;
but the potential relations:
B = curl A; E = -grad phi - dA/dt +
GxB
(with the gauge condition div A = 0). (Maxwell also had the Ohm's law J
= sigma E relation present, but that's not of concern here).
The covariance under a change of velocity, v, is expressed by
d/dt -> d/dt - v.del; del -> del; t -> t; r ->
r + vt
and
B -> B; D -> D; E -> E; H -> H; G -> G + v
In modern notation, E is defined by the relation E = -grad phi - dA/dt;
so that the relation of Maxwell's E, which I'll call E_M, to that used
today is:
E_M = E + G x B,
and the constitutive relations read (in modern notation)
D = epsilon (E + G x B).
The corresponding Galilean-invariant relation for H and B would be
H = mu (B - G x D)
and, in fact, the -G x D term was mentioned in passing in a footnote.
Both C and G (as well as the absence of P) were central to Maxwell's
theory, which also is not widely-known down to the present era.
According to Maxwell, both point and line concentrate sources are
impossible -- he even provided a thought experiement, involving a
conductor that descends to a sharp point) to explain what would happen.
The dielectric property, according to Maxwell, was NOT a property of
materials, but a universal phenomena attributable to all space. In
particular, though the breakdown voltage increase sharply as one
evacuates air from a region of space (something Maxwell explained in
detail and provided numerical estimates on), one cannot conclude from
this that the breakdown would tend toward infinity as the density of
air goes to 0 and (in fact) one has to almost certainly infer that the
breakdown voltage CONTINUE to be finite, even in what "is otherwise
known as a vacuum"; so that one will have vacuum polarization and
breakdown of the vacuum, itself.
In virtue of this property, infinite concentrations on a line or point
of charges will tend to create a screening effect in close vicinity,
with the breakdown of the surrounding medium, leading to a dampening of
the charge down to a finite value; and one thus also speaks of a bare
vs. effective charge (a section in the treatise was devoted to
discussing the notion of effective charges vs. bare charges).
Thus, D represents, not a field + polarization, but the total
polarization, itself, with epsilon_0 E being the contribution by the
vacuum. Hence, there is no separate term for P = D - epsilon_0 E. It is
also for this reason that one finds J + dD/dt combined into a "total
current" C, since now (under this point of view) dD/dt represents an
actual motion of charges brought about by the polarization of the
vacuum, itself.
The term G represents the velocity of the underlying dielectric medium.
At the time the first editions of the treatise were written, the values
quoted for light speed were just short of the "orbital velocity"
threshold of uncertainty; the figures being given as +/- 100 km/second.
So, this point in time was just a few years shy of the reduction of
delta(c) under the 50's, 40's, 30's, etc. where the issue of the
Earth's motion would even be relevant.
Consequently, the whole problem of producing a comprehensive analysis
of the transformation properties of the equations and force law never
received anything more, in the treatise, than scattered, incomplete,
fragmented treatment.
This, at last, gets us to what Einstein actually did, and will help
explain some of the otherwise cryptic remarks in the initial section of
his 1905 paper.
It was this gap, in the treatise's coverage of the subject, of not
fully analysing the transformation properties of the equations that
Einstein was actually reacting to, and which comprised the largest part
of his paper.
In the abstract, the ending passage about no longer needing to assume
an "aether frame" was actually in reference to this distinction between
constitutive relations:
The one with the Aether Frame (G):
D = epsilon (E + G x B); B = mu (H - G x D)
versus the (Poincare') relations
Without the Aether Frame
D = epsilon E; B = mu H.
It was not a repudiation of some vague "Aether Theory", but that
portion of Maxwell's equations, themselves, that leads to the first set
of relations.
When saying "we can do without the Aether hypothesis", it's NOT saying
that "the Aether is no longer necessary because it produces an
equivalent theory to the Aether-free theory", but quite the opposite:
"the Aether is no longer necessary, because it produces the WRONG
theory, which is NOT equivalent (to the Poincare' theory)".
This runs completely opposite to the folk-history of the subject, which
has Einstein eliminating the "Aether" from an "equivalent" version of
Maxwell because it's no longer "necessary".
The two versions of Maxwell, the Galilean/Aether version with Maxwell's
Aether velocity (G) and Poincare', are not equivalent.
The essential contribution was to not only recognize the Poincare'
relations as the primary ones, but to explain how they can come about
as a consequence of the fundamental properties of space and time. Up to
that point in time, the research had been derected toward finding some
way to coax something resembling the latter set of relations out of the
former set, which orginates with Maxwell, which were repudiated the
moment c became known to within a +/- 10 km/second uncertainty or less.
Part of why all this confusion about an "Aether theory" got mixed up in
all this, in fact, because of the tie-in to the other part of Maxwell's
theory concerning vacuum polarization and the universal dielectric.
Though Einstein showed in clear fashion that Maxwell's G is not needed,
he did not in any way, show that the underlying theory of Maxwell
(which is well-grounded on the mere requirement of consistency and
freedom from the point and line infinities) concerning the universal
dielectric, vacuum polarization, and charge screening.
In the process of "throwing out the Aether", history has unwittingly
also thrown out the "Maxwell theory of renormalization", which is more
than 150 years ahead of its time.
This part of Maxwell's treatise survives the transition from classical
to quantum theory, and his account for the elimination of the classical
infinity (which is the one inherited by quantum theory), survives
intact, as well.
Much of it is present as the underpinning to the modern-day theory of
renormalization, but not all of it has been incorporated in modern
theory.
What it implies, among other things, is that (D,H) and (E,B) must
continue to be regarded as separate fields, even though (D,H) may have
a relation to (E,B) in virtue of the dielectric medium. The argument
concerning infinities has equal bearing today, with surprising
consequences that have not yet been fully incorporated:
In the presence of point sources, where rho(r) goes
singular as r -> r_0,
* D(r) goes singular as r -> r_0, since div D = rho.
* The force law, F = rho E, however, must remain
well-defined
* Therefore, E(r) can NOT go singular as r -> r_0,
* Thus, D = epsilon_0 E can NOT hold in close vicinity
to a point-like source.
Either the sources are not point-like, or the constitutive relations
(as well as the property of Poincare' invariance that is linked to it)
break down near point-sources -- or both.
To fully incorporate Maxwell's theory will ultimately entail a more
comprehensive theory, free from the classical infinities when rendered
classically (and free from infinities when cast in quantum theoretic
form), in which (D = epsilon_0 E) has to be sacrificed. In the vicinity
of point-like sources, Poincare' invariance, itself, will break down or
appear to do so as the Poincare' constitutive relations break down.
This phenomenon, in turn, will be directly connected to what in
renormalization theory is called "charge screening" and "vacuum
polarization".
.
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